David J. Green

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Cohomology of group number 2177 of order 128

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General information on the group

• The group has 5 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 5.
• Its center has rank 3.
• It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 5.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 5 and depth 4.
• The depth exceeds the Duflot bound, which is 3.
• The Poincaré series is

  ( − 1) · (t3  +  t  +  1)

  (t  +  1) · (t  −  1)5 · (t2  +  1)

• The a-invariants are -∞,-∞,-∞,-∞,-6,-5. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 4:

1. b_1_0, an element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an element of degree 1
4. b_1_3, an element of degree 1
5. c_1_4, a Duflot regular element of degree 1
6. c_2_13, a Duflot regular element of degree 2
7. b_3_29, an element of degree 3
8. b_3_30, an element of degree 3
9. c_4_59, a Duflot regular element of degree 4

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Ring relations

There are 9 minimal relations of maximal degree 6:

1. b_1_0·b_1_2
2. b_1_1·b_1_2 + b_1_0·b_1_3 + b_1_0·b_1_1
3. b_1_0·b_1_3² + b_1_0·b_1_1·b_1_3
4. b_1_2·b_3_29
5. b_1_0·b_3_30
6. b_1_3·b_3_29 + b_1_1·b_3_30 + b_1_1·b_3_29
7. b_3_29·b_3_30 + c_2_13·b_1_1·b_1_3³ + c_2_13·b_1_1²·b_1_3²
8. b_3_30² + b_1_2·b_1_3²·b_3_30 + b_1_2²·b_1_3·b_3_30 + b_1_2³·b_3_30
      + c_4_59·b_1_2² + c_2_13·b_1_3⁴ + c_2_13·b_1_1²·b_1_3²
9. b_3_29² + b_1_0·b_1_1²·b_3_29 + b_1_0²·b_1_1·b_3_29 + b_1_0⁵·b_1_1
      + c_4_59·b_1_0² + c_2_13·b_1_1²·b_1_3² + c_2_13·b_1_0²·b_1_1²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 7.
• However, the last relation was already found in degree 6 and the last generator in degree 4.
• The following is a filter regular homogeneous system of parameters:
  1. c_1_4, a Duflot regular element of degree 1
  2. c_2_13, a Duflot regular element of degree 2
  3. c_4_59, a Duflot regular element of degree 4
  4. b_1_3² + b_1_2·b_1_3 + b_1_2² + b_1_1·b_1_3 + b_1_1² + b_1_0·b_1_1 + b_1_0², an element of degree 2
  5. b_1_2·b_1_3² + b_1_2²·b_1_3 + b_1_1·b_1_3² + b_1_1²·b_1_3 + b_1_0·b_1_1·b_1_3
        + b_1_0²·b_1_1, an element of degree 3
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 3, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
• We found that there exists some filter regular HSOP formed by the first 3 terms of the above HSOP, together with 2 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. c_1_4 → c_1_0, an element of degree 1
6. c_2_13 → c_1_2², an element of degree 2
7. b_3_29 → 0, an element of degree 3
8. b_3_30 → 0, an element of degree 3
9. c_4_59 → c_1_1⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

1. b_1_0 → c_1_3, an element of degree 1
2. b_1_1 → c_1_4, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → c_1_4, an element of degree 1
5. c_1_4 → c_1_0, an element of degree 1
6. c_2_13 → c_1_2·c_1_3 + c_1_2², an element of degree 2
7. b_3_29 → c_1_3²·c_1_4 + c_1_2·c_1_4² + c_1_2·c_1_3·c_1_4 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
8. b_3_30 → 0, an element of degree 3
9. c_4_59 → c_1_3·c_1_4³ + c_1_3³·c_1_4 + c_1_1·c_1_3·c_1_4² + c_1_1·c_1_3²·c_1_4
      + c_1_1²·c_1_4² + c_1_1²·c_1_3·c_1_4 + c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → c_1_4, an element of degree 1
5. c_1_4 → c_1_0, an element of degree 1
6. c_2_13 → c_1_2², an element of degree 2
7. b_3_29 → c_1_2·c_1_3·c_1_4, an element of degree 3
8. b_3_30 → c_1_2·c_1_4² + c_1_2·c_1_3·c_1_4, an element of degree 3
9. c_4_59 → c_1_2·c_1_3·c_1_4² + c_1_2·c_1_3²·c_1_4 + c_1_2²·c_1_4² + c_1_2²·c_1_3·c_1_4
      + c_1_1·c_1_3·c_1_4² + c_1_1·c_1_3²·c_1_4 + c_1_1²·c_1_4² + c_1_1²·c_1_3·c_1_4
      + c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_1_3 → c_1_4, an element of degree 1
5. c_1_4 → c_1_0, an element of degree 1
6. c_2_13 → c_1_2·c_1_3 + c_1_2², an element of degree 2
7. b_3_29 → 0, an element of degree 3
8. b_3_30 → c_1_2·c_1_4² + c_1_2·c_1_3·c_1_4 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
9. c_4_59 → c_1_2·c_1_3²·c_1_4 + c_1_2²·c_1_4² + c_1_1·c_1_3·c_1_4² + c_1_1·c_1_3²·c_1_4
      + c_1_1·c_1_3³ + c_1_1²·c_1_4² + c_1_1²·c_1_3·c_1_4 + c_1_1⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128